1. Stochastic Facility Location Planning on Networks Subject 8 Christian Lohmann

2. CONTENTS 1.Introduction 2.Explanation of terms 3.1-Median-Problem 4.Stochastic–Queue-Median-Model 5.Conclusion

3. 1. INTRODUCTION How to locate an Ambulance Station in a network? Primary goal of locating an Emergency Facility: Location criteria: Travel-distance to the demand point Travel-time to the demand point Cost function relating both travel-time and travel-distance to the incident In a real-life situation the purpose is to protect and rescue human lives by choosing a location that minimizes the travel-distance or -time to the incident/demand point. Main problem in the model: Expected uncertainty in the field of the locations of incidents. The service-demand point is unknown therefore, we are using probabilities to handle the problem

4. 2. EXPLANATION OF TERMS 54 32 1 l=3 Network: g=0,1 g=0,35 g=0,1 l=1 l=2 l=4 l=2 l=5

5. 2. EXPLANATION OF TERMS The network consists of: 1. NODES (1-5): The nodes represent possible and random demand or customer points. That means that from each of the five nodes of the network an Emergency-call may arise. The aim is to locate the ambulance station, called server/facility, in one of the five nodes. 2. LINKS (l): Link means here the distance between two nodes. In our model the distances are deterministic. For example: l (2,3) = 3 3. CONDITIONAL PROBABILITY (g): Indicates the probability that a service demand/call for service originates from one of the five nodes. The sum of the five probabilities (g) equals one.

6. 3. 1-MEDIAN-PROBLEM Median: Median was founded by Hakimi in 1964 and therefore, it is named: „Hakimi´s 1-Median Problem“ Median is that point of the network that minimizes the weighted sum of distances to the other nodes of the network. Median is a location that optimize minisum criteria: In our underlying network the criterion is to minimize the average travel-distance to the other nodes. In order to find the optimal location for the ambulance-station in our network we make use of the Expected Travel Time and the Second Moment of Service Time. The optimal location is that node that minimizes the Expected Travel Time. We use the stochastic tools to solve the location problem because of the fact that we are confronted with uncertainy regarding to the service demand points. There exist only conditional probabilities to judge from which node of the network may arise a service- demand. From this point of view the location of service demand is stochastic.

7. 3. 1-MEDIAN-PROBLEM Expected Travel Time: t(x) = 1/v. Σ g i. d(x, i) for all i = 1,...,n „v“ stands for the travel velocity „d(x,i)“ means the distance between the facility location „x“ and the random demand point „i“ on the network. In our model the distances between the nodes are deterministic. „g i “ is the conditional probability that a service demand originates from the node „i“ on the network. t(x) is the weighted sum of travel time from fhe facility location „x“ to the demand point „i“

8. 3. 1-MEDIAN-PROBLEM Second Moment of Service Time: s 2 (x) = Σ g i. (d(x, i)/v + (ß-1). d(i, x)/v + α) 2 for all i = 1,...,n „ß“ is a roundtrip travel factor. If ß equals to two, then the travel time to the facility location is the same as the travel time of initial response. „ α“ is the mean of a non-travel-related term, which is independent of the location of the server.

9. 3. 1-MEDIAN-PROBLEM Expected Travel Time and Second Moments of Service Time: Node (Location of Facility) 1 2 3 4 5 t (Expected Travel Time) S 2 (Second Moments of Service Time) 3,25 2,85 3,75 3,15 4,15 81,8 71,0 87,0 79,0 112,6 The optimal location that minimizes the Expected Travel Time and the Second Moment of Service Time is node 2. Node 2 represents „Hakimi´s 1-Median“. In this underlying network the optimal facility location for the Ambulance Station is node 2.

10. 4. STOCHASTIC-QUEUE-MEDIAN-MODEL In a real life situation with a certain probability more than one service demand/call for service arise at the same moment. Queue: Queueing effects arise when one server/facility has to treat more than one service demand/call for service at the same moment. The arrival of service demands from a temporal point of view is accidental The case of congestion of the facility/server is the motivation for dealing with queueing effects. Congestion occurs if the server, at the moment of a second call, is still treating the first service demand. Therefore, the server is unable to handle the second call and it is entered into an infinite capacity queue that is depleted in a FiFo (First-in-First-out)- manner. Hakimi´s 1-median problem is extended by embedding it in a queueing context. The „Stochastic –Queue-Median-Problem“ contains both the Median and the Queueing aspect.

11. 4. STOCHASTIC-QUEUE-MEDIAN-MODEL We develop our model further and take queueing effects into account: The objective now is to minimize the Average Response Time to random demand points: TR (x, λ)= W q + t (x) The average response time (TR) is defined by 1.) x : Server/facility location on the network 2.) λ: Lambda stands for the average Poisson arrival rate of calls for service. Lambda is the stochastic variable because on the one hand the number of calls is absolutely unclear and on the other hand the arrival of service demand from a temporal point of view is uncertain. In this underlying model Stochastic Facility Location Planning is caused and determined by Lambda.

12. 4. STOCHASTIC-QUEUE-MEDIAN-MODEL The Expected Time to Response (TR) associated with a random service demand is the sum of the following two components: 1.) W q : Expresses the Mean-In-Queue-Delay. The value of the mean-in-queue-delay, is said of the in-queue waiting time is determined by the average Poisson arrival rate of calls (λ) and the Service Time. 2.) t(x) : Expresses the Expected Travel Time to the demand. One calculates the Time to Response (TR) for all possible server-/facility-locations on the network and chooses that location that minimizes the Time to Response (TR). TR (x*) < TR (x) The optimal locaton is x*, called Stochastic-Queue-Median

13. 4. STOCHASTIC-QUEUE-MEDIAN-MODEL We are varying values of Lambda in order to state how the optimal facility location changes: Optimal Facility Locations: λ Optimal Location, x* (λ) TR 0 Node 2 2,85 0,01 Node 2 3,23 0,02 x = 0,887 on (a,b) = (2,4) 3,63 0,05 x = 1,568 on (a,b) = (2,4) 5,153 0,06 x = 1,614 on (a,b) = (2,4) 5,893 0,07 x = 1,627 on (a,b) = (2,4) 6,838 0,08 x = 1,609 on (a,b) = (2,4) 8,086 0,09 x = 1,557 on (a,b) = (2,4) 9,809 0,11 x = 1,278 on (a,b) = (2,4) 16,344 0,14 Node 2 83,011

14. 4. STOCHASTIC-QUEUE-MEDIAN-MODEL Expected Travel Time and Second Moments of Service Time: Node (Location of Facility) 1 2 3 4 5 t (Expected Travel Time) S 2 (Second Moments of Service Time) 2,20 2,75 2,25 4,65 4,25 51,0 61,2 45,2 125,2 109,2 g 1 = 0,35; g 2 = 0,1; g 3 = 0,3; g 4 = 0,125; g 5 = 0,125 α = v = 1; ß = 2

15. 4. STOCHASTIC-QUEUE-MEDIAN-MODEL We are varying values of Lambda in order to state how the optimal facility location changes: Optimal Facility Locations: λ Optimal Location, x* (λ) TR 0 Node 1 2,20 0,01 Node 1 2,469 0,015 Node 1 2,616 0,02 Node 3 2,757 0,05 Node 3 3,808 0,06 Node 3 4,273 0,08 Node 3 5,478 0,11 Node 3 8,543 0,13 Node 3 12,558 0,15 Node 3 21,621 0,16 Node 1 32,200

16. 4. STOCHASTIC-QUEUE-MEDIAN-MODEL 8 109 3 742 651 55 3 5 3 54 4 3 4 4 7 6 6 6 3 5 0,13 0,220,20 0,02 0,09 0,07 0,04 0,15 0,01

17. 4. STOCHASTIC-QUEUE-MEDIAN-MODEL Node i t(x) S(x) S 2 (x) 1 7,91 36,68 1348,96 2 7,53 32,30 1062,70 3 7,35 29,12 879,64 4 5,94 28,71 844,75 5 5,58 31,35 990,73 6 7,20 30,97 988,21 7 6,45 27,22 769,36 8 7,56 25,33 690,49 9 8,57 26,34 749,24 10 10,77 22,54 614,60

18. 4. STOCHASTIC-QUEUE-MEDIAN-MODEL We are varying values of Lambda in order to state how the optimal facility location changes: Optimal Facility Locations: λ Optimal Location, x* (λ) TR 0 Node 5 5,58 0,005 Node 4 8,406 0,01 Node 7 11,735 0,015 Node 8 15,912 0,02 Node 8 21,555 0,025 Node 10 28,370 0,03 Node 10 39,241 0,035 Node 10 61,720 0,04 Node 10 135,689 0,0425 Node 10 321,359

19. 4. STOCHASTIC-QUEUE-MEDIAN-MODEL Total Service Time : S(x) = d(x, i)/v + R i + (β –1).d(i, x)/v + W i = α + β. t(x) „α “ is the mean of the non-travel-related service time and equals to one. „ß“ is the roundtrip travel factor and equals to two. Therefore, the travel time does not differ between the way there and back. Time to Response: TR(x, λ) = W q + t(x) = λ S 2 (x)/2(1- λ S(x)) + t(x) „W q “ stands for the expected in-queue delay experienced by a random call.

20. 4. STOCHASTIC-QUEUE-MEDIAN-MODEL Values of Lambda (λ): λ max = 1/S min (x M ) > 0 The maximum allowable value of Lambda is 1/S min > 0. For all λ bigger than λ max the queue is unstable.

21. 5. CONCLUSION We distinguish three different values of the random Poisson arrival rate Lambda in order to summarize the main results: 1.) Small values of Lambda: - Queueing effects are negligible. - The optimal location choice is influenced by the travel time. - The objective is to minimize the expected travel time. - The unique Hakimi-Median results as the optimal solution. 2.) Intermediate values of Lambda: - The congestion aspect becomes more important as the value of λ increases - Numerator of W q (x), „λ.S 2“ becomes the key factor and dominates TR - The mean-queue-delay, is said the „in-queue-waiting time“, is affected by the second moment of service time („S 2“) and dominates the solution.

22. 5. CONCLUSION 3.) Large values of Lambda: - The mean service time S(x) is of primary importance and dominates the solution. - The Stochastic-Queue-Median is again the Hakimi-1-Median. Conclusion:- The optimal location changes. - The stochastic variable λ is responsible for different locations solutions. - As λ changes, changes the optimal facility/server location as well. - The level of system congestion which is caused by the value of λ determines the optimal location.